Spin-Orbit Symmetry: SU(2) to SO(3) and Cubic Branching Restrictions — E8 Intelligence Research

FINDING: SU(2) is the universal double-cover of SO(3), with half-integer spin representations irreducible under SU(2) but projective (double-valued) under SO(3); branching to the octahedral group (cubic symmetry) selects integer-spin irreps compatible with crystallographic point groups. MATH: - Covering map: SU(2) → SO(3), kernel = {±I} ≅ ℤ₂. - Spin-\(j\) representation dimension: \(d = 2j+1\). - Branching rule (octahedral group \(O_h ⊂ SO(3)\)): For integer \(j\): \( ResSO(3)O_h D^j = _Γ n_Γ Γ \), where \(Γ\) are irreps of \(O_h\) (A₁, A₂, E, T₁, T₂). For half-integer \(j\): no single-valued rep of \(O_h\) exists — requires double group \(O_h'\) (order 48), with spinor irreps \(E1/2, E3/2, G3/2\). - Cartan decomposition of su(N) (from arXiv:quant-ph/0603190): uses conjugate partitions and quotient algebras — relevant to branching via Weyl character formulas. CONNECTION: - Octahedral group \(O_h\) has order 48 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com